Beignets: In a medium pot, add the first 5 ingredients and bring to a boil over medium heat. At this point, add in the flour and stir vigorously with a spoon. Continue to cook until the dough becomes a solid mass and begins to pull away from the sides of the pot, about 3 to 5 minutes. Switch off the heat, mix in the vanilla extract and add in the eggs, 1 at a time, incorporating each egg before adding in the next.*
The resulting pastry should be flexible and soft, but firm enough to hold its shape. Set it aside covered with plastic for 5 minutes. When ready to make the pastries, fill a deep skillet or deep-fryer 2/3 full of oil and heat it to 300 to 325 degrees F. Form the dough into doughnut hole shapes. Fry the beignets, in batches, until golden brown, about 3 to 6 minutes being sure not crowd them. Remove them from the oil to a cooling rack.
Caramel: Put the rosemary and water into a small pot and simmer over medium heat until the rosemary is limp, about 3 to 6 minutes. Strain the liquid, discarding the rosemary, and pour it back into the pot. (Chef's Note: If this liquid is not 1/4 cup exactly, add water.) Add in the sugar and corn syrup. Bring the ingredients to a boil over high heat, then reduce the heat to medium and simmer until the syrup thickens. Stir in the heavy cream and butter and switch off the heat. Pour into a serving bowl and reserve for the beignets.
Arrange the beignets on a serving platter and dust with powdered sugar. Serve with the honey and Rosemary Caramel in small bowls.
*Work quickly or else the eggs will curdle.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.